Proximal Causal Inference

Definition

When unmeasured confounding $U$ is present, the causal effect is identified using two types of proxies:

• NCE (treatment-inducing confounding proxy) $Z$: $Z\perp Y\mid U,A,X$
• NCO (outcome-inducing) $W$: $W\perp A\mid U,X$

If the outcome confounding bridge $h$ solves the Fredholm integral equation $E[Y\mid Z,A,X]=E\big[h(W,A,X)\mid Z,A,X\big]$ then the causal effect is identified as a functional of $h$. Completeness (the condition that the proxies sufficiently illuminate $U$) guarantees solvability.

Intuitive Understanding

Even though $U$ cannot be measured, its two “shadows” (proxies) let us triangulate the confounding and recover the effect. The core idea is that, even when a single modality is incomplete, fusion can achieve completeness.

Related Concepts

• Negative Control Outcome · Partial Identification · Sensitivity Analysis · CATE · Efficient Influence Function (semiparametric proximal)

Key Papers

• Miao, Geng & Tchetgen Tchetgen, Biometrika 105(4):987–993, 2018 — nonparametric identification via proxies
• Tchetgen Tchetgen, Ying, Cui, Shi & Miao, “An Introduction to Proximal Causal Inference”, Statistical Science 39(3):375–390, 2024
• Cui, Pu, Shi, Miao & Tchetgen Tchetgen, “Semiparametric Proximal Causal Inference”, JASA 119(546):1348–1359, 2024